Abstract
A search is presented for rare decays of the Z and Higgs bosons to a photon and a J/$\psi$ or a $\psi$(2S) meson, with the charmonium state subsequentially decaying to a pair of muons. The data set corresponds to an integrated luminosity of 123 fb$^{-1}$ of proton-proton collisions at a center-of-mass energy of 13 TeV collected with the CMS detector at the LHC. No evidence for branching fractions of these rare decay channels larger than predicted in the standard model is observed. Upper limits at 95% confidence level are set: $\mathcal{B}$(H $\to$ J/$\psi \gamma$) $\lt$ 2.6 $\times$ 10$^{-4}$, $\mathcal{B}$(H$\to$ $\psi$(2S)$\gamma$) $\lt$ 9.9 $\times$ 10$^{-4}$, $\mathcal{B}$(Z$\to$ J/$\psi$$\gamma$) $\lt$ 0.6 $\times$ 10$^{-6}$, and $\mathcal{B}$(Z$\to$$\psi$(2S)$\gamma$) $\lt$ 1.3 $\times$ 10$^{-6}$. The ratio of the Higgs boson coupling modifiers $\kappa_\mathrm{c}/\kappa_\gamma$ is constrained to be in the interval ($-$157, $+$199) at 95% confidence level. Assuming $\kappa_\gamma=1$, this interval becomes ($-$166, $+$208).